DISTRIBUTION OF THE FILTERED OUTPUT OF A QUADRATIC RECTIFIER COMPUTED BY NUMERICAL CONTOUR INTEGRATION

The numerical quadrature of a Laplace inversion integral along a contour in the complex plane (chosen to economize the number of steps) is used to compute the cumulative distribution of the filtered output of a quadratic rectifier whose input is either narrowband Gaussian noise or Gaussian noise with a low-pass spectral density. The integrand contains the moment-generating function (mgf) of the output expressed in terms of the Fredholm determinant, and the resolvent kernel associated with an integral equation. The resolvent kernel involves the autocovariance function of the input and the impulse response of the output filter. A special case considered is that of the power of a mean-zero Gaussian process averaged over a finite interval when this process has a rational spectral density. The mgf is then expressed as the ratio of certain finite determinants. Using this method, distributions are calculated for low-pass noise with RLC and second- and fourth-order Chebyshev spectral densities. For rational input spectral densities with arbitrary positive output filtering and an arbitrary additive input signal, the mgf is calculated by integrating differential equations of the Kalman-Bucy type.